Yes. There is a special form needed to transport a machine gun, or a short barreled rifle, or a short barreled shotgun out of state. This form is not needed for an AOW, and I don't think it is needed for a silencer (but the memory might be wrong on that one). You need to send in a form 5320. http://test.titleii.com/5320-20.html
There is a block on there for "reason". If moving permanently, you say, "permanent move". If just temporarily, then put "shooting competition". You need to give them an address of where it will be stored while out of state, and the dates when it will be out of state.